Divergence

Definition · applies to vector fields · also called $\nabla \cdot F$, $\operatorname{div} F$

High school

Consider a flow, like air moving in a room. The divergence of the flow measures how much it spreads out from each point (positive) or gathers in toward it (negative).

Undergraduate

Consider a vector field $F = (F_1, \dots, F_n) : \reals^n \to \reals^n$. The divergence of $F$ is the scalar function

\[\nabla \cdot F = \frac{\partial F_1}{\partial x_1} + \dots + \frac{\partial F_n}{\partial x_n}.\]

It measures how much $F$ spreads out from a point.

Properties

• The divergence is the trace of the Jacobian matrix of $F$.
• By the divergence theorem, the integral of $\nabla \cdot F$ over a region equals the flux of $F$ out through its boundary.

Notes

Unrelated to a divergent sequence.

Distinctions involving this term

• Types of derivatives in vector calculus: The Jacobian of a vector field contains all of its first partial derivatives. The divergence (a scalar) and curl (a vector) summarize parts of it. The Laplacian is a second-order operator on scalar functions.

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